A number of the transitive graphs on 16 vertices can be constructed from 2Q3, two copies of the cube. Then add additional edges. For example Q4, the Clebsch graph, and the Shrikande graph can all be constructed
in this fashion. It is possible to chose the vertices of 2Q3 in four
groups of four, and induce K4's on these subsets, etc., to obtain a
number of vertex-transitive graphs.